How Do You Compute a Lipschitz Constant for a Trigonometric Function?

[transmini]

Compute a Lipschitz constant K as in (3.7) $$f(t, y_2)-f(t, y_1)=K(y_2-y_1) \space\space (3.7)$$, and then show that the function f satisfies the Lipschitz condition in the region indicated:

$$f(t, y)=p(t)\cos{y}+q(t)\sin{y},\space {(t, y) | \space |t|\leq 100, |y|<\infty}$$ where p,q are continuous functions on $$-100\leq t \leq 100$$

I honestly have no idea how to even begin this. Other than the definition on Lipschitz continuity (f and df/dy are continuous on the region given) the book being used doesn't really talk about anything Lipschitz.

And just as disclaimer, this is *technically* homework however its nothing turned in or for a grade. Just something for practice.

Any help, especially with at least getting started, is much appreciated.

 

[S.G. Janssens]

I would write (3.7) as
$$
|f(t,y_2) - f(t,y_1)| \le K |y_2 - y_1|
$$
and I assume that $K$ is supposed to not depend on $t$.

Hint: Explain and use the fact that $\tfrac{df}{dy}$ is bounded on the given region, say $\Omega$. Argue that $K := \sup_{(t,y) \in \Omega}{\left|\tfrac{df}{dy}(t,y)\right|}$ can be taken as a Lipschitz constant.

Does this come from a text on ODEs?

 

[fresh_42]

$p(t),q(t)$ are continuous on the compact set $[-100,100]$, aren't they? What am I missing?

Edit: Got it. I've automatically associated a "$\leq$" with Lipschitz, not the actual equality.